1. Technical Field
This invention pertains to determination of the distance from a receiver of a source of electromagnetic or other waves, and to separation of signals from different sources depending on the respective distances. More particularly, it concerns extraction of spectral components so as to exhibit a dependence on the distances of the sources that can then be used for determining the source distance or for separating the signals from individual sources.
2. Brief Description of the Prior Art
Distance-Dependent Scaling of Frequencies
In a first copending application, titled “Passive distance measurement using spectral phase gradients”, filed 2 Jul. 2004, Ser. No. 10/884,353, and incorporated herein by reference, a method has been described for extracting the information of the distance to a source of electromagnetic or other wave signal received by a receiver by causing, at the receiver, the spectrum of the signal to shift in proportion to the distance and independently of the signal modulation and content.
The shift is characterized by a parametric operator H(β) defined as
                                                        H              ⁡                              (                β                )                                      ⁢                                                        ω                ,                r                            〉                                =                                    ⅇ                              ⅈ                ⁢                                                                  ⁢                                  kr                  ⁡                                      (                                          1                      +                                              β                        ⁢                                                                                                  ⁢                                                  r                          /                          c                                                                                      )                                                                        ⁢                                                        ω                                  1                  +                                      β                    ⁢                                                                                  ⁢                                          r                      /                      c                                                                                  〉                                      ,                            (        1        )            where r is the source distance, k and ω respectively denote the wave vector and the angular frequency, c is the wave speed, |ω,r represents the incoming wave state in quantum mechanical notation, and
                              β          =                                                    1                                  ω                  ^                                            ⁢                                                ⅆ                                      ω                    ^                                                                    ⅆ                  t                                                      ≡                                          1                                  k                  ^                                            ⁢                                                ⅆ                                      k                    ^                                                                    ⅆ                  t                                                                    ,                            (        2        )            where t is the observation time and {circumflex over (k)} and {circumflex over (ω)}, the wave vector and angular frequency instantaneously selected at the receiver.
Equation (1) reveals that the shifts would be proportional to the original frequencies, as in the Doppler effect, and thus amount to a scaling of frequencies by the scale factor
                                          z            ⁡                          (                              β                ,                r                            )                                ≡                      δω                          ω              ^                                      =                                            β              ⁢                                                          ⁢              r                        c                    =                                    α              ⁢                                                          ⁢              r              ⁢                                                          ⁢              where              ⁢                                                          ⁢              α                        ≡                          β              /                              c                .                                                                        (        3        )            Unlike the Doppler effect, this scaling is independent of relative motion, depending instead on the relative distance, and is inherently asymmetric, as the causative parameter β is receiver-defined. Equation (2) further reveals that continuous variation of the receiver's instantaneous selection or tuning, represented by the rate of change factor d{circumflex over (k)}/dt, is key to obtaining the shift, which is also described by the resulting orthogonality condition{circumflex over (ω)}|H(β)|ω,t≡∫tei{circumflex over (ω)}Δ(t)ei[krΔ(r)−ωt]dt=∫teikrΔei({circumflex over (ω)}Δ−wt)dt≡eikrΔδ[{circumflex over (ω)}Δ−ω],  (4)wherein the first factor exp[i{circumflex over (ω)}Δ(t)≡{circumflex over (ω)}|H(β) represents the (forward) Fourier transform kernel {circumflex over (ω)}|≡exp(i{circumflex over (ω)}t) modified by varying the instantaneous selection or tuning, with Δ≡Δ(r)=(1+βr/c) and Δ≡Δ(t)=(1+βt), since c=r/t. The modified transform no longer selects ordinary Fourier components exp[i(kr−ωt)], but would clearly accept H−1(β)|ω≡exp[i(kr−ωt/Δ)] as the Δ factors would cancel out. The eigenfunction exp [i(kr−ωt/Δ)] is a spectral component of time-varying frequency, representing the view of a receiver whose very scale of time is instantaneously changing.
Such decomposition, while not of much use in prior art, is legitimately within a receiver's prerogative for analyzing received signals. The eigenfunction is also equivalent to exp[i(krΔ−ωt)], projecting the variation of scale from time to path length. Wavefunctions of varying frequency are known as chirps in radar, but in all art prior to the first copending application, chirps have been expressly generated as in radar and otherwise sought in received signals only from specific chirp sources, such as gravitational waves from a collapsing binary star system. Chirp transforms have been also applied in image processing for their capability for extracting or preserving scale-related features that would be lost in ordinary Fourier methods. The difference over prior art in the first copending application is that the chirps are extracted from arbitrary received waveforms.
Equation (4) implies that each incoming Fourier component ω in effect gets measured at its scaled value ω/Δ. As explained in the first copending application, the mechanism critically depends on the fact that no real signal can be absolutely monochromatic since nonzero spreading of frequency is necessary for energy and information transport. Equation (2) relates the distance-frequency scale factor β to the instantaneous rate of scanning of the received signal spectrum, and the measured amplitude or energy at each selected frequency {circumflex over (ω)} comes from integration, via equation (4), over a nonzero differential interval of this spread around the corresponding source frequency ω. Injection of the shift factor Δ into the forward Fourier kernel exp(i{circumflex over (ω)}t) on the left of equation (4) results from a further relation
                              δω          =                                                                      ∂                  ϕ                                                  ∂                  k                                            ⁢                                                ⅆ                                      k                    ^                                                                    ⅆ                  t                                                      =                                                                                                      ω                      ^                                        ⁢                    β                    ⁢                                                                                  ⁢                    r                                    c                                ⁢                                                                  ⁢                so                ⁢                                                                  ⁢                that                ⁢                                                                  ⁢                ω                            =                                                ω                  ^                                +                δω                                                    ,                            (        5        )            obtained from a first principles consideration of the instantaneous phase of an incoming sinusoidal wave,
                                                                        ⅆ                ϕ                                            ⅆ                t                                      ≡                          -                              ω                ^                                              =                                                    ∂                ϕ                                            ∂                t                                      +                                                            ∂                  ϕ                                                  ∂                  r                                            ⁢                                                ⅆ                  r                                                  ⅆ                  t                                                      +                                                            ∂                  ϕ                                                  ∂                  k                                            ⁢                                                ⅆ                                      k                    ^                                                                    ⅆ                  t                                                                    ,                            (        6        )            in which the first term on the right, ∂φ/∂t≡∂(kr−ωt)/∂t≡−∂(ω[t−r/c])/∂t=−ω, is the intrinsic rate of change of phase of the incoming wave; the second term denotes the Doppler effect of relative motion (˜dr/dt) if any of the source; and the last term accounts for the changing phase contribution due to variation of the instantaneous selection {circumflex over (k)} at the receiver, its first factor representing the spectral phase gradient
                                                        ∂              ϕ                                      ∂              k                                ≡                                    ∂                              (                                                      k                    ⁢                                                                                  ⁢                    r                                    -                                      ω                    ⁢                                                                                  ⁢                    t                                                  )                                                    ∂              k                                      =                  r          .                                    (        7        )            Equation (5) follows from combining equations (6) and (7), for stationary sources, and indicates that the instantaneous selection {circumflex over (ω)} measures the amplitude or energy at ω in the actual incoming spectrum.
As each component of the received spectrum would be scaled independently, the amplitude distribution would be generally preserved. Although the spread of frequencies would be mostly due to modulation in a communication application, the phase contribution of modulation would be equal to a fluctuation (t) of the source location around =r with zero mean deviation, and the modulation thus belongs to the signal part −iωt instead of the space part ikr of the instantaneous phase i(kr−ωt) of the carrier.
Utility of the Distance-Dependent Scaling.
The above mechanism finally provides a way to determine the physical distance to a wave signal source in a fundamental way analogous to determining the direction of a source using a directional antenna or an antenna array.
A second copending application titled “Distance division multiplexing”, Ser. No. 11/069,152, filed 1 Mar. 2005, with a priority date of 24 Aug. 2004 and also incorporated herein by reference, describes a combination of the H operator with conventional filters to separate signals received simultaneously from sources located at different distances, even if lying along the same direction, without regard to their modulation or content.
The result is a fundamental means to selectively receive a desired signal source that can be employed independently of all known techniques of time frequency or code division multiplexing, or in combination with these techniques, thereby making source distance or location a fundamental new dimension on par with time frequency and spread-spectrum coding for multiplexing and demultiplexing. It enables space division multiple access in a truer sense than hitherto applied to the division of solid angles subtended at communication satellite transponders.
Further, as described in a paper titled “Relaxed bandwidth sharing with Space Division Multiplexing”, at the IEEE Wireless Communication and Networking Conference, March 2005, by the present inventor, the result also fundamentally improves over the traditional Shannon capacity bound of communication channels because the bound is based on the assumption that frequency, or equivalently time is the only physical dimension available for discriminating between signal sources and also between signal and noise. Spread-spectrum coding cannot improve this bound because it concerns modulation within the channel, redistributing the use of the spectrum rather than introducing a new dimension. Other parameters have contributed physical space as a multiplexing and multiple access dimension only in a relatively limited sense, notably in the following ways.
In cellular communication, direction can be used for better reuse of channels in neighbouring cells, at the cost of added antenna complexity. Division of the solid angle subtended by the earth's surface at a communication satellite, using directional antennas, has long been used to multiply the total number of transponded channels. Polarization is frequently used to double the number of channels in telemetry, and is now also being used for wireless and cellular communication, as described, for instance, by Juerg Sokat et al in U.S. Pat. No. 5,903,238, issued 11 May 1999. Small antenna arrays are being researched for similarly improving link bandwidths, and this is somewhat inappropriately termed space division multiplexing by some authors, in view of the larger antenna cross-sections and multiplexing of the directional patterns of the antenna elements, albeit only within the point-to-point link and between the link end points.
The prior ideas of using the spatial dimensions for multiplexing thus have been limited to single link or cellular scenarios. They did not enable general use of physical space as a multiplexing dimension that in principle would have permitted discrimination and selection of signals from an unlimited number of sources, and might have obviated time or frequency division, and the base stations networks currently needed for cellular localization of channel allocations.
Prior Methods and their Difficulties.
Based on the equations cited above, the means described in the first copending application for realizing H generally involve continuously varying intervals, of sampling in digital signal processing suitable at long wavelengths, or of diffraction gratings, for use at optical wavelengths. As such these mechanisms would have to be incorporated in the frontend antennae or optical subsystems, which are generally difficult to access and would entail custom design. In the method of the second copending application, for instance, an inverse shift H−1(β)≡H(−β) is needed to restore selected signals back to their original frequency bands, following a first shift by H(β) and band-pass filters to select the signals; the signal samples have had to be interpolated to the corresponding nonuniform sample instants. Similar interpolation could be used also for the first H(β) operator with conventional frontend subsystems. Correctness of inversion is also unobvious because of the reinterpolation, and its empirical verification would always be incomplete as it is physically impossible to test with all possible input signals. A fundamentally different approach seems necessary even to complete the theoretic picture.
There are also practical problems with interpolating samples, namely that every stage of interpolation adds noise due to its finite order and precision. Moreover, each interpolation must ensure an exponential profile of sampling interval variation in order to obtain a uniformly scaled spectrum. This requirement comes from equation (2) due to a correspondence of {circumflex over (k)} to the instantaneous sampling interval τ established in the first copending application. Small deviations from the exponent are to be expected, and would further blur the spectrum. Difficulties of uniformly varying grating intervals, let alone exponentially over time has been already noted in the first copending application as one of the key contributing reasons why such shifts were hitherto unnoticed.
While analogue-to-digital convertors (ADC) routinely provide 8 or 12 bits per sample, corresponding to 256 or 4096 quantization levels, respectively, for audio, similar precision at radio frequencies (RF) is generally unavailable. Interpolating samples quantized at 3 or 9 levels, as in radio telescopes including the Arecibo, could render the data too noisy. The digitization is commonly performed at intermediate frequencies (IF), at which the phase differences are smaller and could be lost in the interpolation noise. Moreover, much ongoing work in radio astronomy concerns the power spectrum at 100 GHz and above, for which the conventional ADC-DFT scheme becomes quite impractical. Since there is little interest in the time-domain signal waveform per se, the method preferred is correlation spectroscopy, wherein an auto-correlation is first computed, whose Fourier transform then directly yields the power spectrum via the Wiener-Khintchine theorem.
Early correlators were all-analogue, comprising two or more tapped delay lines (or “lags”), one set for each channel, two or more channels being fed in pairs in opposite directions and multiplied together at each tap to compactly obtain the autocorrelation function. These correlation outputs are typically integrated for a preset interval before being digitized and input to a DFT. Increasingly, the correlations are performed digitally, which avoids attenuation in the lags that had limited the bandwidth of analogue correlators. As the correlation already contains the power spectrum information, the selection of chirp eigenfunctions must be performed before or within the correlation, i.e. before the sampling for the DFT in the analogue case, so the mechanism of varying the sampling interval is useless in this fairly common case broadly representative of high frequency spectrometry.
Summary of the Problem and Need
The basic difficulty for a fundamental alternative is the requirement that the receiver variation cannot be equivalent to any static nonuniform pattern. Exponential rulings, for instance, can be produced with precision, but as explained in the first copending application, they would merely blur the spectrum instead of scaling, as the rays of different frequencies, diffracted from different regions of the nonuniform grating, would be combined simultaneously. Such a grating could be spun on a transverse axis, so as to instantaneously present exponentially varying grating intervals to the rays, and the blur should then diminish to reveal scaling, but the result would not be different from the methods of the first copending application, and would not adequately address the difficulties identified above.
A need exists, therefore, for a method that can provide the distance-dependent scaling of signal spectra fundamentally without requiring varying of sampling or grating intervals. Such a method could be also more generally suited for optical applications, like correlation spectroscopy, and would be preferable in any case also in digital systems from the perspective of noise.